All states have restrictions about marrying relatives. The degree of separation varies widely. Some allow 1st cousins, some do not.
exogamy
In the US it is legal in 6 states for first cousins to marry, under certain circumstances. See link.
Third cousins can marry in all states of the United States. Abut half the states forbid marriages of first cousins; none forbid marriages of second cousins.
Yes, second cousins can legally get married.
I do not believe that it is illegal for a married man to move in with a woman that is married to a different man. The only law that may apply is if the man decides to marry the woman before either of them get divorced. That is called polygamy, and is illegal in all 50 states.
It is not illegal to date anyone. It is illegal to engage in sexual activity with siblings. This is called incest. While it is typically socially taboo to date or have sex with cousins, it is not illegal in most states.
Your question is a little unclear, but I presume that you meant to ask if cousins could get married. Some states allow it, some do not.
Yes, Utah is one of the few states that all types of cousins (1st, 2nd, etc) can be married. While 1st cousins may be married in Utah there may be restrictions. (age/reproduction)
No a person cannot legally have a wedding ceremony while still married to another. It's called bigamy and is illegal in all States.
Yes, an illegal immigrant from Bulgaria can get married in the United States. An American can choose to marry whomever they want.
Certainly. And in most states you can get married if you have the same grandparents.
About half of the states in the US permit marriage of first cousins.